We prove the Yau-Tian-Donaldson's conjecture for any Q-Fano variety that has a log smooth resolution of singularities such that a negative linear combination of exceptional divisors is relatively ample and the discrepancies of all exceptional divisors are non-positive. In other words, if such a Fano variety is K-polystable, then it admits a Kähler-Einstein metric. This extends the previous result for smooth Fano varieties to this class of singular Q-Fano varieties, which includes all Q-factorial Q-Fano varieties that admit crepant log resolutions.

Abstract. In this paper, we give a new version of the modified Futaki invariant for a test configuration associated to the soliton action on a Fano manifold. Our version will naturally come from toric test configurations defined by Donaldson for toric manifolds. As an application, we show that the modified K-energy is proper for toric invariant Kähler potentials on a toric Fano manifold. IntroductionLet (M, g) be a Fano manifold with a Kähler form ω g ∈ 2πc 1 (M) of g. Denote η(M) to be the linear space of holomorphic vector fields on M. Then by Hodge Theorem, for any X ∈ η(M), there exists a unique smooth complex-valued functionwhere h g is the Ricci potential of g such thatIt was shown there that F X (v) is a holomorphic invariant independent of the choice of g with ω g ∈ 2πc 1 (M), and so it defines an obstruction to the existence of Kähler-Ricci solitons with respect to an element X ∈ η r (M), where η r (M) is the reductive part of η(M). In particular, when. It was also proved by Tian and Zhu that there exists a unique X such thatRecently, by using Ding-Tian's idea of generalizing Futaki invariant [DT], Xiong and Berman, gave a generalization of the modified Futaki invariant F X (·) for any special degeneration associated to X, independently [Xi,Be2]. As a consequence, they both proved that F X (·) is nonnegative if M admits a Kähler-Ricci soliton. Berman also gaves an algebraic formula for F X (·), which depends on weights of the automorphisms group on holomorphic sections of multi-line bundles on the center fibre induced by the test configuration. The purpose of present paper is to define the modified Futaki invariant F X (·) for any test configuration associated to X. Our motivation is inspired by Berman's algebraic formula for special degenerations and is to modify his formula for general test configurations. Then by applying the We note that the above energy argument was used by other people, such as in [JMR, LS, T3, LZ] to study the conical Kähler-Einstein metrics on general Fano manifolds. Theorem 0.1 and Theorem 0.2 will be proved in Section 2-3 and Section 4, respectively.Acknowledgements. The third author would like to thank professor Gang Tian for his interest to the paper and sharing his insight in Kähler geometry.

Let X be a Q-Fano variety and Aut(X) 0 be the identity component of the automorphism group of X. Let G denote a connected reductive subgroup of Aut(X) 0 . We prove that if X is G-uniformly K-stable, then it admits a Kähler-Einstein metric. The converse of this result holds true if G is a maximal torus of Aut(X) 0 , or is equal to Aut(X) 0 itself. These results give (equivariantly uniform) versions of Yau-Tian-Donaldson conjecture for arbitrary singular Fano varieties. A key new ingredient is a valuative criterion for the G-uniform K-stability.

We prove the Yau‐Tian‐Donaldson conjecture for any ℚ‐Fano variety that has a log smooth resolution of singularities such that a negative linear combination of exceptional divisors is relatively ample and the discrepancies of all exceptional divisors are nonpositive. In other words, if such a Fano variety is K‐polystable, then it admits a Kähler‐Einstein metric. This extends the previous result for smooth Fano varieties to this class of singular ℚ‐Fano varieties, which includes all ℚ‐factorial ℚ‐Fano varieties that admit crepant log resolutions. © 2020 Wiley Periodicals LLC

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